A declared finite phase-averaging model reproduces the which-path law for orthogonal components

Adopt the declared two-path preparation and detection model The declared two-path preparation model and its ideal detection bins with path components ${\psi }_1,{\psi }_2\mathrm{\ne }0$ satisfying $⟨{\psi }_1,{\psi }_2⟩=0$ (orthogonal components), and the coherent and which-path detection laws Coherent and which-path detection laws of the declared two-path model. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), as required by the preparation and detection-law suppliers. Fix an integer $n\ge 2$ and declare the finite phase-averaging model: the relative phase takes the $n$ values ${\phi }_k:=\frac{2\pi k}{n}(k=0,1,\dots ,n-1)$ with equal statistical weight $1\mathrm{/}n$, so that the averaged bin law of a Borel set $B\subseteq {\mathbb{R}}^3$ is $\bar{p}(B):=\frac{1}{n}{\sum }_{k=0}^{n-1}{\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}({\phi }_k)}^G(B)\mathrm{.}$ Put $N:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2$ and $w_{\ast }:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2\mathrm{/}N$. Then:

(a) $\mathrm{\parallel }{\psi }_1+e^{i{\phi }_k}{\psi }_2{\mathrm{\parallel }}^2=N$ for every $k$, so every coherent preparation of the declared model has the same normalization $N$;

(b) the averaged cross term vanishes bin by bin: $\frac{1}{n}{\sum }_{k=0}^{n-1}2\mathrm{Re}(e^{-i{\phi }_k}{\int }_B⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}dx)=0;$

(c) consequently $\bar{p}(B)={\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w_{\ast })}^G(B)$ for every Borel set $B$: the declared finite phase-averaging model reproduces the which-path law with the norm weights, while each single phase shows the cross term of Coherent and which-path detection laws of the declared two-path model.

The averaging over the phase is a declared statistical model of the preparation ensemble in the sense of clause (ii) of the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions; the computation asserts no decoherence mechanism, no detector response, no counts and no empirical result, and it does not cover the continuous uniform phase distribution, which would require its own integral model.

• Source scope. Tong supplies the two-slit cross term and discusses which-path monitoring; Binney and Skinner discuss alternative-route amplitudes and spatial averaging over unresolved fringes. Neither cited passage specifies this finite random-phase ensemble. Its hypotheses are declared locally, and its averaged law is proved here using the detection laws and the sum of roots of unity.
• The averaging is not a dynamics. Nothing here evolves the state or removes coherence dynamically; the computation concerns a declared ensemble of preparations whose relative phase is randomized over a finite set. A continuous uniform phase distribution is a different declared model and is not computed here.
• Orthogonality is used. The identification of the averaged law with the which-path law uses $⟨{\psi }_1,{\psi }_2⟩=0$, which makes the normalization independent of the phase. For non-orthogonal components the averaged law need not coincide with the which-path law, and no claim is made about that case.